Properties

Label 2736.2.k.h
Level $2736$
Weight $2$
Character orbit 2736.k
Analytic conductor $21.847$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 912)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{5} -2 \beta q^{7} +O(q^{10})\) \( q + 2 q^{5} -2 \beta q^{7} -\beta q^{11} + 2 \beta q^{13} + 2 q^{17} + ( -1 - 3 \beta ) q^{19} -\beta q^{23} - q^{25} -5 \beta q^{29} + 6 q^{31} -4 \beta q^{35} + 8 \beta q^{37} -3 \beta q^{41} -5 \beta q^{47} - q^{49} -7 \beta q^{53} -2 \beta q^{55} + 8 q^{59} -8 q^{61} + 4 \beta q^{65} -2 q^{67} + 8 q^{71} -4 q^{77} -8 q^{79} -9 \beta q^{83} + 4 q^{85} + 7 \beta q^{89} + 8 q^{91} + ( -2 - 6 \beta ) q^{95} -2 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} + O(q^{10}) \) \( 2q + 4q^{5} + 4q^{17} - 2q^{19} - 2q^{25} + 12q^{31} - 2q^{49} + 16q^{59} - 16q^{61} - 4q^{67} + 16q^{71} - 8q^{77} - 16q^{79} + 8q^{85} + 16q^{91} - 4q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
1.41421i
1.41421i
0 0 0 2.00000 0 2.82843i 0 0 0
2431.2 0 0 0 2.00000 0 2.82843i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.k.h 2
3.b odd 2 1 912.2.k.a 2
4.b odd 2 1 2736.2.k.i 2
12.b even 2 1 912.2.k.d yes 2
19.b odd 2 1 2736.2.k.i 2
24.f even 2 1 3648.2.k.c 2
24.h odd 2 1 3648.2.k.f 2
57.d even 2 1 912.2.k.d yes 2
76.d even 2 1 inner 2736.2.k.h 2
228.b odd 2 1 912.2.k.a 2
456.l odd 2 1 3648.2.k.f 2
456.p even 2 1 3648.2.k.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.k.a 2 3.b odd 2 1
912.2.k.a 2 228.b odd 2 1
912.2.k.d yes 2 12.b even 2 1
912.2.k.d yes 2 57.d even 2 1
2736.2.k.h 2 1.a even 1 1 trivial
2736.2.k.h 2 76.d even 2 1 inner
2736.2.k.i 2 4.b odd 2 1
2736.2.k.i 2 19.b odd 2 1
3648.2.k.c 2 24.f even 2 1
3648.2.k.c 2 456.p even 2 1
3648.2.k.f 2 24.h odd 2 1
3648.2.k.f 2 456.l odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5} - 2 \)
\( T_{7}^{2} + 8 \)
\( T_{11}^{2} + 2 \)
\( T_{31} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -2 + T )^{2} \)
$7$ \( 8 + T^{2} \)
$11$ \( 2 + T^{2} \)
$13$ \( 8 + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 19 + 2 T + T^{2} \)
$23$ \( 2 + T^{2} \)
$29$ \( 50 + T^{2} \)
$31$ \( ( -6 + T )^{2} \)
$37$ \( 128 + T^{2} \)
$41$ \( 18 + T^{2} \)
$43$ \( T^{2} \)
$47$ \( 50 + T^{2} \)
$53$ \( 98 + T^{2} \)
$59$ \( ( -8 + T )^{2} \)
$61$ \( ( 8 + T )^{2} \)
$67$ \( ( 2 + T )^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 162 + T^{2} \)
$89$ \( 98 + T^{2} \)
$97$ \( 8 + T^{2} \)
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